Q: If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

$$P(A \cap B) = P(A) \times P(B) = 0.2$$

Q: A bag contains 5 red balls and 7 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

$$P(\text{both red}) = \frac{5}{12} \times \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Q: If $$X is a $$binomial random variable with $$n = 5$$ and $$p = 0.6$$, what is the variance of $$X$$?

$$\$$sigma^2$$ = n p (1-p) = 5 \times 0.6 \times 0.4 = 1.2$$

Question 1

Suppose events $$A$$ and $$B$$ are mutually exclusive. If $$P(A) = 0.3$$ and $$P(B) = 0.5$$, what is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) = 0.8$$

Question 2

A fair die is rolled. What is the probability of getting either a 2 or a 5?

Accepted Answer

$$P(2 \cup 5) = \frac{2}{6} = \frac{1}{3}$$

Question 3

If $$P(A) = 0.4$$, $$P(B) = 0.5$$, and $$A$$ and $$B$$ are independent, what is $$P(A \cap B)$$?

Accepted Answer

$$P(A \cap B) = P(A) 	imes P(B) = 0.2$$

Question 4

A bag contains 5 red balls and 7 blue balls. If two balls are drawn without replacement, what is the probability that both are red?

Accepted Answer

$$P(	ext{both red}) = \frac{5}{12} 	imes \frac{4}{11} = \frac{20}{132} = \frac{5}{33}$$

Question 5

Given $$P(A) = 0.6$$, $$P(B) = 0.5$$, and $$P(A \cap B) = 0.3$$, what is $$P(A \mid B)$$?

Accepted Answer

$$P(A \mid B) = \frac{P(A \cap B)}{P(B)} = \frac{0.3}{0.5} = 0.6$$

Question 6

How many ways can you arrange 4 different books on a shelf?

Accepted Answer

$$n! = 4! = 24$$

Question 7

A committee of 3 is to be chosen from 8 people. How many different committees are possible?

Accepted Answer

$$\binom{8}{3} = \frac{8!}{3!5!} = 56$$

Question 8

A binomial random variable $$X$$ has $$n = 10$$ and $$p = 0.2. $$What is the probability of exactly 3 successes?

Accepted Answer

$$P(X = 3) = \binom{10}{3} (0.2)^3$ (0.8)^7$ \approx 0.201$$

Question 9

For a Poisson process with average rate $$\mu = 3$$ per hour, what is the probability of observing exactly 2 events in one hour?

Accepted Answer

$$P(X = 2) = \frac{3^2$ e$$^{-3}$$}{2!} \approx 0.224$$

Question 10

The mean of a discrete random variable $$X$$ with probability mass function $$P(x) is $$given by:

Accepted Answer

$$\mu = \sum x \cdot P(x)$$

Question 11

If $$X is a $$binomial random variable with $$n = 5$$ and $$p = 0.6$$, what is the variance of $$X$$?

Accepted Answer

$$\$$sigma^2$$ = n p (1-p) = 5 	imes 0.6 	imes 0.4 = 1.2$$

Question 12

For a standard normal distribution, what is the $$z-$$score for $$x = 12$$, $$\mu = 10$$, $$\sigma = 2$$?

Accepted Answer

$$z = \frac{x - \mu}{\sigma} = \frac{12 - 10}{2} = 1$$

Question 13

A continuous random variable $$X$$ has a uniform distribution on $$[0, 5]. $$What is the probability that $$X \u003c 3$$?

Accepted Answer

$$P(X \u003c 3) = \frac{3}{5} = 0.6$$

Question 14

If $$P(A) = 0.7$$ and $$P(B) = 0.4$$, and $$A$$ and $$B$$ are not mutually exclusive, $$P(A \cap B) = 0.2. $$What is $$P(A \cup B)$$?

Accepted Answer

$$P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.7 + 0.4 - 0.2 = 0.9$$

Question 15

A hospital records the number of patients arriving per hour as a Poisson process with $$\mu = 4. $$What is the probability that exactly 0 patients arrive in an hour?

Accepted Answer

$$P(X = 0) = \frac{4^0$ e$$^{-4}$$}{0!} = e$$^{-4}$$ \approx 0.018$$

Question 16

Which of the following is true for a valid probability mass function $$P(x)$$?

Accepted Answer

$$\sum P(x) = 1$$

Question 17

A standard normal table gives $$P(Z  1.96)$$?

Accepted Answer

$$P(Z \u003e 1.96) = 1 - 0.975 = 0.025$$

Question 18

A researcher wants to know the probability that a randomly selected student scores between $$z = -1$$ and $$z = 1 on a $$standard normal distribution. What is this probability?

Accepted Answer

$$P(-1 \u003c Z \u003c 1) = P(Z \u003c 1) - P(Z \u003c -1) = 0.8413 - 0.1587 = 0.6826$$

Probability and Discrete/Continuous Distributions: Exam 2 Study Notes

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